The barycenter method on singular spaces

Abstract

Compact convex cores with totally geodesic boundary are proven to uniquely minimize volume over all hyperbolic 3-manifolds in the same homotopy class. This solves a conjecture in Kleinian groups concerning acylindrical 3-manifolds. Closed hyperbolic manifolds are proven to uniquely minimize volume over all compact hyperbolic cone-manifolds in the same homotopy class with cone angles $\le 2\pi$. Closed hyperbolic manifolds are proven to minimize volume over all compact Alexandrov spaces with curvature bounded below by $-1$ in the same homotopy class. A version of the Besson–Courtois–Gallot theorem is proven for $n$-manifolds with boundary. The proofs extend the techniques of Besson–Courtois–Gallot.